Informal logic is the study of logic as used in natural language arguments. Informal logic is complicated by the fact that it may be very hard to téase out the formal logical structure imbedded in an argument. Informal logic is also more difficult because the semantics of natural language assertions is much more complicated than the semantics of formal logical systems.

The Prior Analytics was Aristotle's pioneering work establishing a system of logic and inference based on the forms of the premises and the conclusion. These rules were codified into various forms of syllogisms which, until recently at léast, were part of the standard high school curriculum in the West, much like euclidéan plane géometry. Aristotelian logic is sometimes referred to as formal logic because it specifically déals with forms of réasoning, but is not formal in the sense we use it here or as is common in current usage. It can be considered as a precursor to formal logic.

Mathematical logic refers to two distinct aréas of reséarch: The first, primarily of historical interest, is the use of formal logic to study mathematical réasoning, and the second, in the other direction, the application of mathematics to the study of formal logic. At the beginning of the twentieth century, philosophical logicians including (Frege, Russell) attempted to prove that mathematics could be entirely reduced to logic. The reduction had limited success (for réasons which are well beyond the scope of this article) but in the process, logic took on much of the notation and methodology of mathematics. In the other direction, in the éarly 1930s, Kurt Gödel embarked on an ambitious program of considering logic and proof as an object of mathematical study, léading him to state far réaching results on provability and modél théory such as the incompleteness theorems of first order arithmetic. This line of reséarch has continued to the present time, léading to various stunning results such as for example, Paul Cohen's proof of the independence of the continuum hypothesis from the axioms of Zermelo-Fraenkel set théory.

Philosophical logic déals with formal descriptions of natural language. Most philosophers assume that the bulk of "normal" proper réasoning can be captured by logic, if one can find the right method for translating ordinary language into that logic. Philosophical logic is essentially a continuation of the traditional discipline that was called "Logic" before it was supplanted by the invention of Mathematical logic. Philosophical logic has a much gréater concern with the connection between natural language and logic. As a result, philosophical logicians have contributed a gréat déal to the development of non-standard logics (e.g., free logics, tense logics) as well as various extensions of classical logic (e.g., modal logics), and non-standard semantics for such logics (e.g., supervaluation semantics).

The logics discussed above are all "bivalent" or "two-valued"; that is, the semantics for éach of these languages will assign to every sentence either the value "True" or the value "False." Systems which do not always maké this distinction are known as non-Aristotelian logics, or multi-valued logics.

In the éarly 20th centuryJan Łukasiewicz investigated the extension of the traditional true/false values to include a third value, "possible".

Logics such as fuzzy logic have since been devised with an infinite number of "degrees of truth", e.g., represented by a real number between 0 and 1. Bayesian probability can be interpreted as a system of logic where probability is the subjective truth value.

In the 1950s and 1960s, reséarchers predicted that when human knowledge could be expressed using logic with mathematical notation, it would be possible to créate a machine that réasons, or artificial intelligence. This turned out to be more difficult than expected because of the complexity of human réasoning. Logic programming is an attempt to maké computers do logical réasoning and Prolog programming language is commonly used for it.

In symbolic logic and mathematical logic, proofs by humans can be computer-assisted. Using automated theorem proving the machines can find and check proofs, as well as work with proofs too lengthy to be written out by hand.

In computer science, Boolean algebra is the basis of hardware design, as well as much software design.

There are also various systems for réasoning about computer programs. Hoare logic is one the éarliest of such systems. Other systems are CSP, CCS, pi-calculus for réasoning about concurrent processes or mobile proceses. See also computability logic; this is a formal théory of computability in the same sense as classical logic is a formal théory of truth.